The Problem:
It's given that $f(z)$ is an entire function on $\mathbb{C}$ and that there exist $M > 0$ and $A > 0$ such that $|f(x+iy)|\le\frac{Ae^{2\pi M|y|}}{1+x^2}$ for all $x,y \in \mathbb{R}$. The task is to prove that for all $\xi \in R$ with $|\xi|>M$, $\int_{-\infty}^{\infty} f(x)e^{−2\pi i\xi x}dx=0$.
My thoughts so far:
So my initial thought was that I have to cleverly choose an entire function and a closed path in $\mathbb{C}$ to evaluate the chosen function over. Given that the line integral around closed paths for an entire function is $0$, I can theoretically show that the contribution of the line integral along the real line is $0$ by showing that the contribution from every other part of the line integral is $0$. I've been playing with different paths and functions, but haven't been successful yet.
The other thing I noticed is that if
$|f(x+iy)|\le\frac{Ae^{2\pi M|y|}}{1+x^2}$
then that implies that
$e^{-2 \pi i \xi (x+iy)}|f(x+iy)|\le\frac{Ae^{2\pi(M|y|-i\xi(x+iy))}}{1+x^2}$
which certainly seems important since the left hand side is the integrand that I want if I let $y$ be $0$ (which it is on the real axis). It doesn't seem that this bound on the function is very useful though, since the bound doesn't go to $0$ except when $x$ is large and $y$ is around $0$, and so it doesn't seem to tell me anything useful about the function closer to the origin.
In all of this, I haven't really appealed to the requirement that $|\xi|$ be larger than $M$, which I'm sure is important.
A push in the right direction would be appreciated.